Issolvi frac{3sqrt{5}}{1+2sqrt{5}}

$$\frac { 3 \sqrt { 5 } } { 1 + 2 \sqrt { 5 } }$$

Evalwa

$$\frac{30-3\sqrt{5}}{19}\approx 1.225884004$$

Passi tas-Soluzzjonijiet

Irrazzjonalizza d-denominatur tal-\frac{3\sqrt{5}}{1+2\sqrt{5}} billi timmultiplika in-numeratur u d-denominatur mill-1-2\sqrt{5}.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{\left(1+2\sqrt{5}\right)\left(1-2\sqrt{5}\right)}$$

Ikkunsidra li \left(1+2\sqrt{5}\right)\left(1-2\sqrt{5}\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1^{2}-\left(2\sqrt{5}\right)^{2}}$$

Ikkalkula 1 bil-power ta' 2 u tikseb 1.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-\left(2\sqrt{5}\right)^{2}}$$

Espandi \left(2\sqrt{5}\right)^{2}.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-2^{2}\left(\sqrt{5}\right)^{2}}$$

Ikkalkula 2 bil-power ta' 2 u tikseb 4.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-4\left(\sqrt{5}\right)^{2}}$$

Il-kwadrat ta' \sqrt{5} huwa 5.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-4\times 5}$$

Immultiplika 4 u 5 biex tikseb 20.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{1-20}$$

Naqqas 20 minn 1 biex tikseb -19.

$$\frac{3\sqrt{5}\left(1-2\sqrt{5}\right)}{-19}$$

Uża l-propjetà distributtiva biex timmultiplika 3\sqrt{5} b'1-2\sqrt{5}.

$$\frac{3\sqrt{5}-6\left(\sqrt{5}\right)^{2}}{-19}$$

Il-kwadrat ta' \sqrt{5} huwa 5.

$$\frac{3\sqrt{5}-6\times 5}{-19}$$

Immultiplika -6 u 5 biex tikseb -30.

$$\frac{3\sqrt{5}-30}{-19}$$